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Localization in Wireless Sensor NetworksRoudy Dagher, Roberto Quilez
To cite this version:Roudy Dagher, Roberto Quilez. Localization in Wireless Sensor Networks. Nathalie Mitton andDavid Simplot-Ryl. Wireless Sensor and Robot Networks From Topology Control to Communica-tion Aspects, Worldscientific, pp.203-247, 2014, 978-981-4551-33-5. <10.1142/9789814551342_0009>.<hal-00926928>
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
Chapter 9
Localization in Wireless Sensor
Networks
Roudy Dagher1,2 and Roberto Quilez2
1 Etineo, France, 2 Inria Lille – Nord Europe, France
Abstract With the proliferation of Wireless Sensor Networks (WSN)
applications, knowing the node current location have become a crucial re-
quirement. Location awareness enables various applications from object
tracking to event monitoring, and also supports core network services such
as: routing, topology control, coverage, boundary detection and clustering.
Therefore, WSN localization have become an important area that attracted
significant research interest. In the most common case, position related
parameters are first extracted from the received measurements, and then
used in a second step for estimating the position of the tracked node by
means of a specific algorithm. From this perspective, this chapter is in-
tended to provide an overview of the major localization techniques, in order
to provide the reader with the necessary inputs to quickly understand the
state-of-the-art and/or apply these techniques to localization problems such
as robot networks. We first review the most common measurement tech-
niques, and study their theoretical accuracy limits in terms of Cramer-Rao
lower bounds. Secondly, we classify the main localization algorithms, taking
those measurements as input in order to provide an estimated position of
the tracked node(s).
203
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
204 Wireless Sensor and Robot Networks
9.1 Introduction
Recent technological advances in micro-electronics, digital electronics and
wireless communication, have made possible the development of low-cost,
low-power, multi-functional and highly integrated sensor nodes that are
able to communicate in a wireless ad-hoc fashion over short distances [3].
These tiny nodes, typically equipped with processing, sensing, power man-
agement and communication capabilities collaborate to form a Wireless
Sensor Network (WSN). Sensed data is typically sent over the network, in
a multi-hop manner, to a control center either directly or via a base sta-
tion/sink. The main constraints in such networks are the limited amount
of energy and computing resources of the nodes.
With the significant development and deployment of WSN, associating
the sensed data with its physical location becomes a crucial requirement.
Knowing the node’s location enables a myriad of location-based applica-
tions such as object tracking, environment monitoring, intrusion detection,
and habitat monitoring [73] [25]. Location estimation also supports core
network services such as: routing, topology control, coverage, boundary
detection and clustering [43].
Localization is defined as the process of obtaining a node location with
respect to a set of known reference positions. It is also referred to as location
estimation or positioning. Nodes at reference positions are called anchor
nodes1, and nodes with unknown positions are called tracked nodes2. Based
on reference positions of a few anchor nodes in the network, and inter-
node measurements such as range and connectivity, localization algorithms
estimate the position of a tracked node in the network. Depending on
targeted applications, the coordinate system may be global or local (e.g.
habitat monitoring).
In the most common case, position related parameters are first extracted
from the received measurements, and then used in a second step for esti-
mating the position of the tracked node by means of a specific approach:
fingerprinting, geometric or statistical methods [20]. The used technique
highly depends on the application’s requirements and challenges:
• Environment
The environment where a WSN is deployed may be challenging, as
localization performance is affected by multipath and non-line-of-sight
1also referred to as reference node, beacon device, base station, etc.2also referred to as non-anchor node, target node, blindfolded device, mobile station
etc.
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
Localization in Wireless Sensor Networks 205
(NLOS) propagation. Environment variability is typically due to: pres-
ence of obstacles, metallic environments acting as wave guides, inter-
ference, etc.
• Complexity
In the context of WSN, nodes are typically battery-powered with lim-
ited computing power and memory. Therefore, it may not be feasible
to implement complex localization algorithms. However, in some cases
the base station may have advanced computing capabilities and act as
a localization server for the application.
• Accuracy
Coarse-grained accuracy of several meters may be sufficient for patient
tracking inside a hospital, and may be addressed by simple low-cost
Zigbee-based solutions [17]. Conversely, fine-grained accuracy usually
requires specialized hardware such as Ultra-wideband (UWB) [22].
• Scalability
Scalability of a localization algorithm determines how well it accom-
modates as the number of nodes increases and the coverage area is
expanded. This metric is very important in dense networks.
• Latency
Depending on the tracked object dynamics, the latency to determine its
location might be a big concern. It should be considered with respect
to other layers such as the Medium Access Control (MAC) layer for
channel access latency.
• Dependability
The system should be able to keep operating even if some anchor nodes
are faulty. This is referred to as system fault tolerance. In [51] a
WSN localization system with error detection/correction is presented.
Another issue to consider is network lifetime, mainly in the case of
battery-powered nodes.
In brief, the choice of a sensor network localization technique often involves
a trade-off among the above-listed constraints in order to suit the require-
ments of the targeted application(s). In essence, these challenges make
localization in wireless sensor networks unique and intriguing. This chap-
ter is intended to provide an overview of the major techniques that have
been widely used for WSNs localization system. Based on the referenced
material, a special effort has been made to broadly classify the different
localization aspects in order to provide a starting block for this topic. The
remainder of this chapter is organized as depicted in Fig. 9.1. First section
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
206 Wireless Sensor and Robot Networks
WSN
Localization
Measurement
Techniques
Methodologies
& Algorithms
Fig. 9.1 Chapter overview.
presents the measurement techniques and their theoretical accuracy limits
in terms of Cramer-Rao lower bounds. The second section covers the lo-
calization theory, strategies and algorithms taking those measurements as
input in order to provide an estimated position of the tracked node(s).
9.2 Measurement Techniques
Position related parameters estimation is the first step of WSN localiza-
tion. This estimation often relies on physical measurements, depending on
the available hardware capabilities. On the other hand, network related
measurements such as hop count, or neighborhood information can lead to
coarse-grained localization that may be sufficient in dense networks. Fig-
ure 9.2 gives an overview of these measurement techniques. It is the type of
measurements employed and the corresponding precision that fundamen-
tally determine the estimation accuracy of a localization system and the
localization algorithm being implemented by this system.
In the case of physical measurements, a result of estimation theory
can be used to bound the localization error: the Cramer-Rao lower bound
(CRLB) [30, 82]. This theoretical bound gives the best performance that
can be achieved by an unbiased location estimator. If θ is an unbiased
estimator of an unknown parameter θ, then its covariance matrix Cov(θ)
is bounded by the CRLB as
Cov(θ) ≥n
− E⇥
rθ(rθ ln f(X|θ))⇤
o−1
(9.1)
where X is the random observation vector with probability density function
f(X|θ)), E[·] indicates the expected value, and rθ is the gradient operator
with respect to θ. Note that the CRLB is independent of the estimation
method, and only depends on the statistical model of the observations.
Therefore, the CRLB can serve as a benchmark for localization algorithms.
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
Localization in Wireless Sensor Networks 207
Position Related
Measurements
Physical
Angle Distance
Signal Strength Time Delay Phase Difference
Network
Hop Count
Neighborhood
AreaAoA
RSSI ToA
TDoA
NFER
RIM
Fig. 9.2 Measurement techniques overview.
In the case where θ is scalar, the CRLB in Eq. (9.1) becomes
σ2θ≥ 1
−Eh
∂2 ln f(X|θ)∂θ2
i =1
−R
R
∂2 ln f(X|θ)∂θ2 f(X|θ)dX
· (9.2)
9.2.1 Physical measurements
Physical measurements can be broadly classified into three categories ac-
cording to the measurement type: angle measurements, distance related
measurements and network related measurements.
9.2.1.1 Angle measurements
The angle or bearing relative to reference nodes is measured by estimating
the angle of arrival (AoA) parameter between the tracked node and ref-
erence nodes. Given the angle measurements, the location of the tracked
node may be determined by triangulation3 [67]. The AoA measurement
is commonly made available by the use of directional antennas or antenna
arrays4, by measuring the phase difference between the signal received by
adjacent antenna elements ∆Φ = 2π∆sinαλ with ∆ the inter-element spac-
ing of the Na elements antenna array, and λ the wavelength. In order to
3The use of triangulation to estimate distances goes back to antiquity: Thales similar
triangles to estimate the height of the pyramids, distances to ships at sea as seen froma cliff, etc.4Another technique uses receiver antenna’s amplitude response [47].
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
208 Wireless Sensor and Robot Networks
•
Reference
Node
•©Tracked
Node
α
(a)
0
⌧1
⌧2
⌧Na
⌧
∆
x
y
α
(b)
Fig. 9.3 AoA measurement with antenna array. (a) AoA definition. (b) Antenna Array.
understand the accuracy limit with this type of measurements, consider
a uniform antenna as in Fig. 9.3. Under the assumption that the signal,
with effective bandwidth β, arrives at the speed of light c at each antenna
element via a single path, with the same signal to noise ratio (SNR) for all
elements, the CRLB for the variance of an unbiased AoA estimate α can
be expressed by [22]:
p
V ar{α} ≥p3cp
2πpSNRβ
p
Na (N2a − 1)∆ cosα
· (9.3)
Equation (9.3) states that the AoA accuracy increases with the SNR, effec-
tive bandwidth, and the array size. Finally, the best accuracy is obtained
when the signal direction and the antenna line are perpendicular i.e., α = 0.
For a detailed overview on AoA measurements, refer to [47].
9.2.1.2 Distance measurements
By using the distance of the tracked node to several reference nodes, the
position of the tracked node can be computed using the multilateration
method [73].
In order to estimate the distance, several ranging techniques have been
developed [47]. Among them, the Received Signal Strength (RSS) based,
and the time based ranging are the most popular. A less popular tech-
nique, is the Near Field Electromagnetic Ranging (NFER) that exploits
specific near-field properties of radio waves for ranging purposes. Yet an-
other less adopted technique, the Radio Interferometric Positioning (RIPS)
that applies interferometry to radio waves in WSN.
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Localization in Wireless Sensor Networks 209
d1
d2
d3
(a)
d1
d2
d3
d4
d5
d6
(b)
Fig. 9.4 Distance measurements and multilateration (Ranging circles). (a) Trilatera-tion. (b) Multilateration.
Received Signal Strength This technique is based on the received sig-
nal strength indicator (RSSI), a standard feature found in most of the
current off-the-shelf devices. The most typical RSS systems are based on
propagation loss equations [66], given by the following log-normal shadow-
ing model5
Pr(d)[dBm] = P0(d0)[dBm]− 10nplog10d
d0+Xσ (9.4)
where:
- Pr(d)[dBm] : the received power in dB milliwatts at distance d from
the transmitter.
- P0(d0)[dBm] : the reference power in dB milliwatts at a reference dis-
tance d0 from the transmitter.
- np : the path loss exponent that measures the rate at which the received
signal strength decays with distance. Example of values: 2.0 in free
space, 1.6 to 1.8 inside a building [66].
- Xσ : a zero-mean normal variable, with standard deviation σ, that
accounts the shadowing effect. Example of σ values: 0 dB in free
space, and 5.8 dB inside a building [66].
Note that many other models have also been proposed in the literature [66],
but the log-normal model is the most popular one due to its simplicity.5P [mW ] = 10P [dBm]/10 and P [dBm] = 10 log10(P [mW ]/1mW ).
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
210 Wireless Sensor and Robot Networks
From the channel model in Eq. (9.4), we can derive the distribution of the
RSS-based range measurements
d = d · 10Xσ
10np = 10log
10(d)+ Xσ
10np = e(ln 10)·[log
10(d)+ Xσ
10np]= e
(ln d)+ ln 10
10npXσ ·(9.5)
Therefore RSS-based range measurements are distributed according to a
log-normal distribution
d ⇠ lnN (ln d,σ ln 10
10np)· (9.6)
Which yields to the unbiased range estimator from the RSS measurements
[47]:
d = d0
✓
Pr(d)[mW ]
P0(d0)[mW ]
◆−1/np
e− σ2
2η2n2p ,with η =
10
ln 10· (9.7)
The CRLB of an unbiased range estimator is derived in [64]q
V ar{d} ≥ (ln 10)σ
10np· d. (9.8)
Equation (9.8) shows that the ranging accuracy depends on the channel
parameters np and σ, that are environment dependent. Also it deteriorates
as the distance between the transmitter and the receiver increases. Thus,
in order to maintain the estimation error of less than δd, the tracked node
has to be within the range of [64]
r0 =10
ln 10· σ
np· δd. (9.9)
Finally, the result in Eq. (9.8) is generalized by [59], in the case of N
reference nodes and one tracked node in the 2D case:q
V ar{d} ≥ (ln 10)σ
10np·
0
@
v
u
u
t
PNi=1 d
−2i
PN−1i=1
PNj=i+1
⇣
d⊥ijdij
d2
id2
j
⌘
1
A (9.10)
where di is the distance between reference and tracked nodes, dij is the
distance between reference nodes i and j, and d⊥ij j is the shortest distance
from the tracked node to the line segment connecting nodes i and j. This
result highlights the impact of the geometric distribution of the reference
nodes on the localization accuracy.
Propagation Time Distance between neighboring nodes can be esti-
mated using propagation time measurements. Namely, Time of Arrival
(ToA) methods are used when reference and tracked nodes are synchro-
nized, or Time Difference of Arrival (TDoA) that only requires synchro-
nization between reference nodes. Time delay measurements commonly
use generalized cross-correlation or matched filter receivers [22, 35].
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
Localization in Wireless Sensor Networks 211
Time of Arrival (ToA) There are two categories of ToA-based
distance measurements: one-way propagation time, and round-trip propa-
gation time measurements (cf. Fig. 9.5). The one-way propagation time
Node A Node B
tA
tB
τ
| |
(a)
Node A Node B
tA1
tB1
τ
tB2− tB1
tB2
tA2
τ
delay
| |
(b)
Fig. 9.5 Propagation time. (a) One way. (b) Round trip.
measurements measure the difference between sending time at transmitter
node (tA) and receiving time at receiver node (tB). This delay is related
to the inter-node distance by τ = d/c. The main drawback of this tech-
nique, is that the local time of the transmitter and the receiver should be
accurately synchronized. Assuming Line Of Sight (LOS) conditions and
Gaussian noise at receiver level, the CRLB for one-way propagation time
ranging is given by [45, 64]q
V ar{d} ≥ c
2p2π
· 1β· 1p
SNR. (9.11)
Equation (9.11) shows that, unlike RSS-based distance estimation, the ac-
curacy can be improved by increasing the effective signal bandwidth β
and/or the SNR. However, in practice, the transmitter-receiver synchro-
nization requirement increases the cost and the complexity to the system.
In order to cope with synchronization constraint, the round-trip prop-
agation time measurements measure the difference between sending time
tA1at transmitter node and the time when the signal is echoed back by the
receiver node tA2. That is
round-trip = 2 · τ = (tA2− tA1
)− (tB2− tB1
).
Note that synchronization between nodes A and B is no longer required
since the same clock is used to compute the round-trip propagation time.
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
212 Wireless Sensor and Robot Networks
In practice, the internal delay for node B to echo the signal, should be
considered with care in order to avoid jitter introducing meters of errors
on distance measurements. For instance, a hardware implementation with
a priori calibration is a good option.
Assuming LOS conditions, Gaussian noise at receiver level, and no
changes in the bandwidth or SNR conditions, the CRLB for round-trip
propagation time ranging is given by [45]q
V ar{d} ≥ 1
2
✓
c
2p2π
· 1β· 1p
SNR
◆
. (9.12)
Equation (9.12) shows that the ranging accuracy for round-trip propagation
time is twice better than the one-way scenario.
With recent advances in radio technology, the UWB signals are used for
accurate time-based ranging, even in challenging environments, at short dis-
tances with very low energy levels [22, 23]. Nanotron Technologies Gmbh
have designed the Chirp Spread Spectrum (CSS) modulation technique [24]
to cope with multi-path effects, operating in the 2.44 GHz band, with ap-
proximately 60MHz effective bandwidth. In order to avoid synchronization
and eliminate clock drift and offset, Nanotron Technologies Gmbh have also
developed SDS-TWR6, an extension of the round-trip propagation time
ranging presented earlier. The CSS modulation confers to SDS-TWR its
robustness to multi-path effects. Furthermore, time-based ranging have
been standardized with release of the 802.15.4a standard, that adopted
both UWB and CSS signaling [29, 70].
Finally, other time-based techniques have been developed, such as the
Cricket system [63] that combines RF communication and ultra-sound
ranging. The idea behind the cricket system is to take benefit from the
fact that, in the air, the speed of the sound is much smaller than the
speed of the light. The radio link is only used to synchronize the ultra-
sound microphones. Other techniques use audible sound for time of arrival
measurements such as Beepbeep [60] and Beep [46]. Finally, the Lighthouse
approach uses laser beams to measure the distance between tiny dust nodes
and the lighthouse, a modified base station device [68].
Time Difference of Arrival (TDoA) Another type of distance-
based measurements is the Time Difference of Arrival (TDoA), that consists
of measuring the difference between the arrival times between two signals
traveling between the target and the tracked nodes. For illustrating the
TDoA principle, consider the case of one tracked node and two reference6Symetrical Double-Sided Two Way Ranging
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Localization in Wireless Sensor Networks 213
nodes. In this case, the position of the tracked node is on a hyperbola,
with foci the reference nodes, as illustrated in Fig. 9.6. With more than
two reference nodes, the estimated position is at the intersection of the
hyperbolas whose foci are at the locations of the reference nodes pairs.
•r1
•r2
•©
d1
d2
(a)
•r1
•r2
Hyperbola from reference
nodes r1 and r2
•r3
Hyperbola from reference
nodes r2 and r3
•©
(b)
Fig. 9.6 TDOA measurements. (a) With two reference nodes. (b) With three referencenodes.
One approach for measuring TDoA is to first measure the ToA for each
signal between each (reference node, tracked node) pairs as follows
⇢
τ1 = τ1 + offset1
τ2 = τ2 + offset2.
(9.13)
(9.14)
Since the reference nodes are synchronized, offset1 = offset2 which leads to
the TDoA estimate
τTDoA = τ1 − τ2 =d1 − d2
c. (9.15)
The corresponding CRLB expression can be deduced from the ToA case,
which shows that its accuracy increases with effective bandwidth and/or
SNR [23]. It is also proved in [64] that TDoA approach cannot perform
better than ToA approach in terms of accuracy limits.
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
214 Wireless Sensor and Robot Networks
Finally, another approach for measuring TDoA is the generalized cross
correlation method, given by
τTDoA = arg maxτ
∣
∣
∣
∣
∣
Z T
0
r1(t)r2(t+ τ)dt
∣
∣
∣
∣
∣
(9.16)
where ri(t) is the signal between the tracked node and the ith reference
node, and T the observation interval. Refer to [22] and [47] for more details
and discussions.
Phase Difference Measurements The position of the tracked node
can also be estimated using phase difference measurements. The below pre-
sented techniques exploit fundamental laws of physics to determine ranging
information. Namely, Near field electromagnetic ranging (NFER) exploits
near field phase behavior discovered by Heinrich Hertz [75], and Radio
Interferometric Measurements (RIM) is inspired by the interferometric po-
sitioning in the optical regime developed thirty years earlier [14, 48].
Near Field Electromagnetic Ranging (NFER) Near field elec-
tromagnetic ranging (NFER) uses the near field phase relationship of elec-
tric and magnetic components to infer range estimate [74, 76].
x
y
z
r
Tracked
Node•©
φ
θ
∆ l
−
+
(a)
r(λ )
φ ◦
0.1
−
0.2
−
0.3
−
0.4
−
0.5
−
0.6
−0
10−
20−
30−
40−
50−
60−
70−
80−
90−
-10−
-20−
-30−
-40−
-50−
-60−
∆Φ◦
Φ◦
H
Φ◦
E
(b)
Fig. 9.7 Near field electromagnetic ranging (NFER). (a) Spherical coordinates, Hertziandipole. (b) Phase difference versus range.
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
Localization in Wireless Sensor Networks 215
Consider a Hertzian dipole of length ∆l, carrying a uniform current,
I = I0 cos(ωt) as in Fig. 9.7. In spherical coordinates (r, θ, φ), the electric
field component Eθ and magnetic field component Hφ are given by [32, 75]
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
Eθ =I0∆l sin θ
4πω✏0
β
r2cos(!t− βr) +
✓
1
r3− β2
r
◆
sin(!t− βr)
]
Hφ =I0∆l sin ✓
4⇡!
1
r2cos(!t− βr)− β
rsin(!t− βr)
]
(9.17)
(9.18)
where β = ωc = 2π
λ , c is the speed of light, λ is the wavelength, and ✏ is the
permittivity of free space. From Eq. (9.17) and Eq. (9.18), we can compute
the phase difference between Eθ and Hφ, which leads to [32, 75]
∆φ = ΦE − ΦH = cot−1⇣! r
c− c
! r
⌘
− cot−1 ! r
c. (9.19)
The plot of ∆φ in Fig. 9.7 clearly shows a direct mapping between ∆φ and
the range r. By using the relationship cot(a− b) = − 1+cot a cot bcot a−cot b , the range
can be related to the phase by [76]:
r =λ
2⇡3
p
cot∆φ. (9.20)
The CRLB of NFER is derived in [32] for both ranging and 2D-
localization. For the ranging case, the CRLB is given by
q
V ar{d} = b
✓
c6 + d6!6
3 c3d2!3
◆
(9.21)
with:
b2 =U2
U3 + V 2(1− U), U =
SNRE + SNRH
2, and V =
SNRE − SNRH
2.
Equation (9.21) shows that accuracy is dependent on real location, the
used frequency, and the signal-to-noise ratio. In the 2D case, the CRLB
additionally shows the impact of the geometrical conditioning, and that the
optimal accuracy cannot be achieved by using only one frequency. Kim et
al. have presented a scheme using multiple frequencies in order to improve
accuracy [31]. In the case of the NFER, CRLB as derived in [32] becomes
a tool for choosing the optimal frequency for a given coverage area.
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
216 Wireless Sensor and Robot Networks
Radio Interferometric Measurements (RIM) Radio Interfero-
metric Ranging, exploits radio frequency interference of two waves emitted
from two reference nodes at slightly different frequencies, in order to obtain
the ranging information for localization by measuring the relative phase off-
set. In [48] authors describe the Radio-Interferometric Positioning (RIPS)
localization method and provide a mean for measuring the phase difference
indirectly using RSSI. In order to illustrate the principle of RIPS, consider
three reference nodes A,B,C and one tracked node D as in Fig. 9.8.
Let A and B transmit at the same time, two unmodulated sine waves
at two close frequencies fa,fb. The resulting interference is a beat signal
with a beat frequency |fa − fb|. The reference node C and the tracked node
D, acting as receivers, will receive the beat signal with a phase difference
depending on the distance difference between the quartet (A,B,C,D)
∆Φ =2π
λ(dAD − dBD − dBC − dAC) mod 2π (9.22)
where λ = c(fa+fb)/2
, and dXY is the euclidean distance between nodes X
and Y . Which yields to the definition of the q-range measurement, defined
as
qABCD = dAD − dBD − dBC − dAC . (9.23)
Note that a single RIPS measurement given by Eq. (9.23), places the tracked
point on a hyperbola branch (cf. Fig. 9.9) whose foci are the transmitter
pair (A,B). The relative phase offset in Eq. (9.22) can then be rearranged
as follows [37]
∆φ =∆Φ
2πλ = qABCD mod λ. (9.24)
Note that several measurements are required at different transmit fre-
quency pairs (fa, fb) in order to resolve the phase ambiguity (due to
mod 2π), therefore the q-range can be found by solving the following system
of n equations
∆φi = qABCD mod λi. (9.25)
In order to find the position of the tracked node, consider the case where
the transmitter pair is (A,C) and the receiver pair (B,D). The associated
q-range is
qACBD = dAD − dCD − dBC − dAB . (9.26)
The position of the tracked node can be obtained by solving the system
of equations obtained from Eq. (9.26) and Eq. (9.26). Geometrically, this
corresponds to the intersections of the two ranging hyperbolas.
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Localization in Wireless Sensor Networks 217
RIPS trilateration is discussed in detail in [37]. For a complete review
of the interferometry in WSN refer to [71].
In [84] authors present an analytical study of the impact of RIPS mea-
surement noise on the localization error. It appears that the localization
error is small if the tracked node is located inside the triangle 4ABC,
and generally increases at a steady exponential rate as the tracked node
moves away from the triangle, unless it is close to LABC . Where LABC is
the union of the three lines representing the extensions of the sides of the
triangle 4ABC, but not including the interiors of the edges of 4ABC.
•A
•B
•C
•©D
dAC
dAD
dBC
dBD
∆Φ
Fig. 9.8 RIPS measurement technique.
9.2.2 Network related measurements
Network connectivity measurements are possibly the simplest measure-
ments. The tracked node location can be inferred by analyzing its neigh-
boring reference nodes in terms of connectivity, radio coverage area, and
neighborhood proximity [43]. This kind of measurements are very cost
effective and straightforward in large-scale networks.
In connectivity measurements, a node measures the number of nodes in
its transmission range. This measurement defines a proximity constraint
between these two nodes, which can be exploited for localization [9, 15]. For
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
218 Wireless Sensor and Robot Networks
•A
•B
•©D
•C
(a)
•A
•B
Hyperbola with (A,B)transmitters
•C
Hyperbola with (A,C)transmitters
•©D
(b)
Fig. 9.9 RIPS localization rounds. (a) First Round. (b) Second Round.
(x1,y1)
(x2,y2)(x3,y3)
Known Position
Unknown Position
−Connection Constraint
Fig. 9.10 Graph illustrating connectivity constraints.
instance, when a tracked node detects three neighboring reference nodes,
it can assume to be close to these nodes and estimate its location as the
centroid of the three reference nodes [26].
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Localization in Wireless Sensor Networks 219
9.3 Localization Theory and Algorithms
In this section, we give a brief introduction to some fundamental theories in
sensor network localization, and a set of major sensor network localization
algorithms are discussed.
The objective behind a positioning methodology is to determinate the
location information of a number of nodes. Location information can be
interpreted as any form of location indicator such as exact location, the
deployment area or the location distribution. As we have seen in Sec. 9.2,
parameters extracted from signals traveling between the nodes will allow
to establish pairwise spatial relationships (angle, distance or proximity).
Localization algorithms will take those parameters as inputs to estimate
the position of the target nodes according to a certain strategy.
Localization Methods
Centralized
Comput. One-Hop Mapping
Distributed
Anchor-based
Range-based
Geometric Statistic
Range-free
Area Hops Neighb.
Anchor-free
Trilat.
Triang.
Circular
Hyperb.
Parametr.
Non-par.
APIT
ROC
DV-H Centroid
SDP
MDS
Lightho. Cooper.
Spring
Coord.
Stitching
Fine−Grained Coarse−Grained
Fig. 9.11 Location methods overview.
There exist well organized surveys in the literature that propose dif-
ferent classifications of the localization systems based on different criteria
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
220 Wireless Sensor and Robot Networks
at different hierarchical levels such as the number of hops, the presence
of reference nodes (anchors) or the computational organization [4, 18, 20,
47, 56, 58, 73, 83]. The purpose of this section is not to provide definitive
and exhaustive taxonomy of the different localization methods. Instead,
this section is intended to present a comprehensive introduction to some of
today’s most popular localization methods. A special effort has been made
to conciliate approaches from different authors to render a general schema.
In the Fig. 9.11, a set of the most representative localization methods are
displayed within some of the implementation choices associated to them.
We have first categorized the localization algorithms into two main
types; namely centralized and distributed, based on the direct dependency
on a centralized resource.
9.3.1 Centralized methods
Centralized localization methods present a direct dependency on a cen-
tralized resource. This resource can be some information previously col-
lected (mapping), a central machine with powerful computational capabil-
ities (centralized computing) or some kind of one hop location reference
(landmark, satellite. . . ) providing this centralized service.
9.3.1.1 Centralized computing
Centralized computing basically migrates inter-node ranging information
and connectivity data to a sufficiently powerful central base station to be
processed. All other nodes in the network only gather the location related
information, such as RSS, and send it to the central processing location.
The base station calculates the estimated location of all the tracked nodes
and communicates it back if requested.
The advantage of centralized processing is minimizing the required capa-
bilities (e.g., processing power and memory space) of the nodes, excepting
the central location processing node. This benefit however comes with a
communication cost, creating high traffic levels and increasing latency since
all the nodes must communicate with a single central receiver to determine
their location. The high level of traffic can cause bottlenecks in the network
and limit the location update rate. The latency and traffic problems get
worse increasing the size of the network. Therefore the centralized pro-
cessing method is more suitable for small network or a network where the
location update rate is low.
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Localization in Wireless Sensor Networks 221
Two representative proposals in this category are Semidefinite Program-
ming (SDP) techniques [15] and Multi-Dimensional Scaling (MDS) [78].
Semidefinite Programming (SDP) Many localization problems can
be formulated as a convex optimization problem. They can be solved us-
ing linear and semidefinite programming (SDP) techniques [15]. SDP is a
generalization of linear programming and has the following form:
Minimize cTx
Subject to: F (x) = F0 + x1F1 + . . .+ xNFN 0
Ax < B
Fk = FTk
(9.27)
where x = [x1y1, x2y2, . . . , xmym, xm+1ym+1, . . . , xNyN ]T . The first m en-
tries are fixed and correspond to the reference nodes positions and the
remaining n −m positions are computed by the algorithm. The objective
is to find a possible position for each target node when a the position of
a set of reference nodes is given. Proximity constraints imposed by known
connections can be represented as linear matrix inequalities (LMIs). In the
case of nodes communicating within a perfect circle, the estimated region
is convex and can not be described by linear equations but as an LMI. For
a maximum radio range Rmax, if two nodes, with positions xi and xj , are
in communication, their separation must be less than Rmax, i.e., it exists
a proximity constraint between them. This can be represented as a radial
constraint and expressed as a LMI: kxj − xik Rmax.
The advantage of this method is that it is simple to model hardware
that provides ranges or angles and simple connectivity. SDP simply finds
the intersection of the constrains. There are efficient computational meth-
ods available for most of convex programming problems. However, this
estimation methodology requires centralized computation. To solve the op-
timization problem, each node must report its connectivity to a central
computer. This approach also requires to handle large data structures and
lacks of scalability because of its complexity. The relevant operation for
radial data is O(n3) where n is the number of convex constraints.
Multi-Dimensional Scaling (MDS-MAP) Multi-dimensional Scaling
(MDS) [78] is often used as part of information visualization techniques for
exploring similarities or dissimilarities. It displays the structure of distance-
like data as a geometric picture.
The typical goal of MDS is to create a configuration of n points in one,
two or three dimensions. Only the inter-point distances are known. MDS
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222 Wireless Sensor and Robot Networks
enables the reconstruction of the relative positions of the point based on
the pairwise distances.
Typical procedure of MDS involves three stages:
• First, compute the shortest path between all pairs of nodes. The short-
est path distances are used to construct the distance matrix for MDS.
The (i, j) entry represents the distance along the shortest path from the
node i to j. If only connectivity information is available, that distance
will be the number of hops. However, it can also incorporate distance
information between neighboring nodes when it is available.
• In a second stage, classical MDS is applied to the distance matrix to
obtain estimated relative node positions.
• Finally , the relative positions are transformed to absolute positions
with the help of some number of fixed anchor nodes.
The strength of MDS-MAP is that it can be used when there are few or
no anchor nodes. It can use both connectivity and distance measurement
ranging techniques and provides both absolute and relative positioning.
However, the main problem with MDS is its poor asymptotic performance,
which is O(n3).
More detailed work based on MDS can be found in [1].
9.3.1.2 One-Hop positioning
This kind of positioning methods require line of sight (LOS), i.e., direct
contact, between the reference position, the landmark, and the node to
locate. The most representative solution of this class is the GPS, where
receiver has to have a clear line of sight to the satellites to operate.
An interesting approach to estimate distance between an optical receiver
and transmitter is the Lighthouse [68] location system. It is a laser-based
solution which allows nodes to autonomously estimate their location with
high accuracy without additional infrastructure components besides a base
station device. The transmitter consists on a parallel optical beam rotating
at a constant speed. The receiver is equipped with an optical sensor and a
clock. Measuring the time it sees the beam (tbeam), the distance (d), from
the base station can be calculated when the rotational speed (or the time
it takes for a complete rotation, (tturn) and the width of the beam (b), are
known:
d =b
2 sin(πtbeam/tturn). (9.28)
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Localization in Wireless Sensor Networks 223
As a result we have a simple ranging system where all the potential positions
of the receiver form a cylinder with radius d centered at the lighthouse rota-
tion axis. Using three such lighthouses in different placements, the location
of the node can be inferred with trilateration principles. Nevertheless, the
original proposal aimed to have a unique base station. For that purpose,
distances are measured in the three axes of the space using mutually per-
pendicular rotation axes in single positioning device. A major advantage
of this system is that the optical receiver can be very small in cost and
size. However the transmitter may be large and expensive and the LOS
requirement remains a big handicap in many practical cases.
9.3.1.3 Profiling techniques
In Sec. 9.2 we have seen different ways to estimate distances between nodes.
Localization algorithms can then be applied to these distances to obtain the
estimated position of the tracked node. Nevertheless, wireless sensor net-
work environments, and specially indoor environments, are often compli-
cated to model and their model parameters determination is also a difficult
task. Such a challenging scenario can be overcome using another approach,
namely profiling-based techniques [5] [36, 61, 69].
The main idea behind these localization techniques, also referred to as
mapping or fingerprinting, is to determine a regression scheme based on
a set of training data and then to estimate the position of a given node
according to that regression function. They work by first constructing a
kind of map of the signal parameters behavior for each anchor node over a
coverage area. In addition to anchor nodes, a collection of n sample points
with a priori chosen positions must be defined to collect that training data.
At each location, li = (xi, yi)T , a vector of signal parametersmi is obtained.
Typically the mij entry corresponds to the value of the signal strength
from/at the anchor j when the anchor node is at location li although other
signal parameters may be used.
These training data can then be expressed as:
τ = {(m1, l1), (m2, l2), . . . , (mn, ln)} . (9.29)
For the training set given in Eq. (9.29), a position estimation rule must then
be determined, i.e., a pattern matching algorithm or regression function to
estimate the location l of a given target node based on a parameter vector
m related to the target node. Some common mapping techniques used in
location estimation include k-nearest neighbors (k-NN) estimation, support
vector regression (SVR) and neural networks [16, 21, 39, 49, 55]. As in [20]
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
224 Wireless Sensor and Robot Networks
we develop k-NN in order to provide an intuition of a simple mapping-based
position estimation.
k-nearest neighbors (k-NN) In its simplest version 1-NN determines
the estimated position of a target node at the location lj in the training set
τ that has the associated vector mj with the shortest Euclidean distance
to the measured parameter vector m:
j = argi∈1,...,n min k m−mi k . (9.30)
In general, k-NN determines the position of the target node with the help of
the k parameters vectors in τ that have the smallest distances to the given
parameter vector m. The position l is then estimated by the weighted sum
of the positions corresponding to those k nearest parameter vectors:
l =k
X
i=1
wi(m)li, (9.31)
where wi is the weighting factor associated to the ith reference location.
Various weights can be used as studied in [49]. For instance, in the uniform
weighted scheme, the sample mean of position is used:
l =1
k
kX
i=1
li. (9.32)
The main advantage of mapping techniques is that they have a certain
degree of inherent robustness. They can provide very accurate position es-
timation in challenging environments with multipath and non-line of sight
propagation. However, the main disadvantage is the requirement that the
training database should be large enough and representative of the current
environment for accurate position estimation. The database should be up-
dated frequently enough so that channel characteristics in the training and
position estimation phases do not differ significantly. Such an update re-
quirement can be very costly for positioning systems operation in dynamic
environments, such an outdoor positioning system.
9.3.2 Distributed algorithms
One way to overcome the traffic bottleneck of the centralized processing
method is to divide the network into sections and allocate a node capable
of executing the positioning algorithm to each section. An alternative ap-
proach, is to distribute the location-estimation task among almost all the
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Localization in Wireless Sensor Networks 225
nodes in the network. In this way, there is no centralized location process-
ing node and each node determines its own location by communication only
with nearby anchor nodes and potentially other tracked nodes. In a fully
distributed processing method, all the nodes must satisfy certain process-
ing capabilities and memory space requirements. One of the advantages of
distributed processing is relatively uniform packet traffic, which makes it
easy to expand the traffic of the network.
Distributed algorithms can be classified according to whether they
use pre-configured reference positions (anchor-based vs anchor-free) or the
granularity of the measured employed, i.e., whether they make some kind
of range (distance or angle) correlation (range-based vs range-free).
9.3.2.1 Anchor-based techniques
Anchor-based algorithms assume that a certain number of nodes, referred
to as anchors or beacons, know their own position through manual con-
figuration or an external positioning system such as GPS. Tracked nodes’
location is then determined by referring to that reference positions with the
help of inter-sensor measurements such us the ones we have seen in Sec. 9.2.
Depending on the measurement techniques employed, anchor-based al-
gorithms can be classified [43] from fine-grained to coarse-grained into sev-
eral categories such as: location, distance, angle, area, hop-count and neigh-
borhood (see Fig. 9.11). This classification allows us to broadly distinguish
localization algorithms between range-based and range-free [26]. Range-
based approaches rely on signal features such as signal strength, time of
flight or angle of arrival for calculating relative distances or angles. In
contrast, range-free methods do not try to estimate direct point-to-point
distance from the received signal parameters; they use topological informa-
tion (connectivity, signal comparison . . . ) rather than ranging.
Range-based localization techniques Range-based localization ap-
plies geometric techniques to estimate the position of a target node. It
uses a set of position related parameters in a number of reference nodes
to describe geometric figures that are supposed to intersect in the point of
interest.
Geometric methods As we have seen in Sec. 9.2, some measure-
ments parameters can define a geometric figure of uncertainly around the
anchor node. The position of the target node can be estimated as the inter-
section of those figures. The received signal strength or the time of arrival
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
226 Wireless Sensor and Robot Networks
of the signal determine a circumference by translating that parameter to
physical distance. Using trilateration the estimated location for the target
node is given by the intersection of three circles from three different refer-
ences, (xi, yi) (see Fig. 9.12(a)). In the case of the time difference of arrival,
the geometric figure corresponds to an hyperbola and an intersection point
can also result as an estimation. On the other hand, the angle of arrival
measure defines a straight line passing through both, the target and the
reference nodes In this case, two parameters are enough to calculate the
estimated position via triangulation.
d1
d2
d3
•(x1,y1)
•(x2,y2)
•(x3,y3)
•©(x, y)
(a)
•(x1,y1)
•(x2,y2)
•(x3,y3)
•©(x, y)
(b)
Fig. 9.12 Location estimation using Trilateration. (a) Ideal case. (b) Including range
error.
Unfortunately, in a practical implementation, the ranging measurements
contain noises. The presence of fading and shadowing may lead this method
to produce no results at all. Error in distance estimation can prevent the
bearing lines to have a common intersection point (see Fig. 9.12(b)). Thus,
an optimization algorithm must be applied to choose an estimated loca-
tion according to some criteria. Two sample algorithms are the circular
and the hyperbolic positioning algorithms. The former optimizes directly
the error associated to distances while the latter makes distance-difference
optimization.
Circular positioning algorithm The Circular Positioning algo-
rithm [40] adopts the criterion of minimizing the sum square error ε. This
error can be expressed as:
ε =
nX
i=1
⇣
p
(xi − x)2 + (yi − y)2 − di
⌘2
, (9.33)
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Localization in Wireless Sensor Networks 227
where (xi, yi) is the position of each reference node. The position (x, y)
of the node that minimizes that error can be calculated using the steepest
descend method defined by:
x
y
]
k+1
=
x
y
]
k
− α
"
δεδxδεδy
#
x=xk,y=yk
. (9.34)
This method requires an initial location to begin the iteration, which can
be the midpoint of the reference positions under consideration.
Hyperbolic positioning algorithm The Hyperbolic Positioning al-
gorithm [40] also referred to as Linear Least Squared errors (LLS) [18] does
not minimize directly the sum of the squared errors of the erroneous dis-
tance estimations to reference positions as in the previous case. Instead it
minimizes a linear function of it by subtracting two distances estimations
i.e., it minimizes the sum of the distance to the hyperbolas resulting from
the subtraction.
Considering n reference positions we can write the distance estimations
(di=1...n) to the target node as:
d2i = (x− xi)2 + (y − yi)
2. (9.35)
To solve the previous system of equations a linearization is performed by
subtracting the location of the first reference from all other equations [54].
The resulting system of equations can be expressed in the form Ax = b as:2
6
6
4
2x1 − 2x2 2y1 − 2y22x1 − 2x3 2y1 − 2y3
. . . . . .
2x1 − 2xn 2y1 − 2yn
3
7
7
5
x
y
]
=
2
6
6
4
d22 − d21 + x21 − x2
2 + y21 − y22d23 − d21 + x2
1 − x23 + y21 − y23
. . .
d2n − d21 + x21 − x2
n + y21 − y2n
3
7
7
5
. (9.36)
Therefore, the estimated position of the target node can be calculated as
the least squares solution of this equation given by:
x
y
]
= (ATA)−1(AT b). (9.37)
Both, circular and hyperbolic algorithms, give the same weight to the
different distance estimations. Nevertheless, as we have seen in Sec. 9.2,
measurements such as received signal strength (RSS) do not depend linearly
on the distance between the nodes. From Eq. (9.4) it can be deduced that
the same error in the RSS measurement will produce larger errors in the
distance estimation if the distance between the nodes is higher. That is, the
accuracy of the distance estimations depends on the distance itself. The
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
228 Wireless Sensor and Robot Networks
use of weighted techniques to improve the accuracy of the hyperbolic and
circular positioning algorithms respectively has been proposed [80]. They
give more weight to those measurements corresponding to short distances,
which accuracy is expected to be greater.
Statistical location techniques Unlike the geometric techniques,
the statistical approach presents a theoretical framework for position esti-
mation for multiple measurement parameters with or without the presence
or noise.
In order to formulate this generic framework, consider the following
model for each of the N estimated parameters:
zi = fi(x, y) + ηi, (9.38)
where ηi is the noise at the corresponding estimation and fi(x, y) is the real
value of the signal parameter at the position (x, y).
As we have seen in Sec. 9.2 for ToA/RSS, AoA and TDoA, fi(x, y) can
be expressed as:
fi(x, y) =
8
>
>
>
>
>
<
>
>
>
>
>
:
p
(x− xi)2 + (y − yi)2 ToA/RSS
tan−1⇣
(y−yi)(x−xi)
⌘
AoA
p
(x− xi)2 + (y − yi)2 −p
(x− x0)2 + (y − y0)2 TDoA(9.39)
In the case that the probability density function of the noise η is known
for a set of parameters, parametric approaches such as Bayesian and Max-
imum Likelihood (ML) can be used. Those techniques are studied in detail
in [20]. In the absence of that information non-parametric methods must
be used. Actually, profiling techniques, such as k-NN, SVR and neural net-
works approaches referred in Sec. 9.3.1.3, are examples of non-parametric
estimators since they do not make any assumption concerning the density
of probability function of the noise.
Range-free localization techniques Cost and hardware limitations
in wireless sensor nodes often prevent the use of range-based localization
schemes. For some applications coarse accuracy is sufficient and range-free
solutions have been revealed as a valid cost-effective alternative. Range-
free based solutions do not try to estimate absolute distances among nodes
using any signal feature such as signal strength, angle of arrival or time or
flight. They, alternatively, use coverage range, i.e., connectivity, or com-
parative features between signals. Three approaches can be distinguished
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Localization in Wireless Sensor Networks 229
according to the granurality of the measurement employed: area, hop-count
and neighborhood based.
Area based Signals coming from beacon nodes can define cover-
age areas described by geometric shapes. The area based location estima-
tion method will compute the intersection of those coverage areas and will
give the centroid of this region as the resulting location estimation for the
tracked node. For instance, if a tracked node receives a signal from another
anchor node, a circular region, centered in that anchor node and radius its
maximum coverage distance, is delimited. When several reference nodes can
be listened, the overlapped area of those circles will determine an estimated
location for the tracked node (see Fig. 9.13(a)). This can be extended to
other scenarios. For example when angular sector can be determined for
the incoming signal from the beacon nodes or when lower coverage bounds
are also available to describe different geometric figures (see Fig. 9.13(b)).
More detailed work about localization based on connectivity-induced con-
straints can be found in [15].
•
••©
(a)
• ••• •• •
•©
(b)
Fig. 9.13 Area measurements. (a) Circles overlapping. (b) Sectors overlapping.
One popular area-based range-free location estimation scheme is
APIT [26]. The APIT algorithm isolates the environment into triangles
(see Fig. 9.14). The vertexes of these triangular regions are anchors nodes
that the tracked node can hear. The presence inside or outside those tri-
angular regions allows to narrow down the area in which the tracked could
reside. The estimated position is obtained from the centroid of the area
provided by the intersection of the reference triangles that contains the
tracked node.
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
230 Wireless Sensor and Robot Networks
•©•
•
•
•
•
•
•
•
•
•
••
•
•
•
•
•
•
Fig. 9.14 APIT algorithm overview.
Another interesting approach is the Ring Overlapping Circles
(ROC) [42] algorithm (see Fig. 9.15). Each anchor broadcasts beacon mes-
sages that will be received by both, neighboring anchors and the tracked
node. By comparing the received signal strength by those anchors to the
one received by the tracked node, the region where the tracked node lies
within can be determined (in light gray in Fig. 9.15). This ring area is
centered in the beacon anchor node and has as higher bound a circle with
radius equal to the distance to the anchor which received signal strength
is immediately inferior to the one received by the tracked node. The lower
bound of the ring is delimited by a circle with radius equal to the distance to
the anchor node with received signal strength immediately superior. The
process is repeated by each anchor node resulting in several overlapping
rings. Finally, the center of gravity of the overlapped area (in dark gray in
Fig. 9.15) is reported as estimated position.
Two key assumptions are made by the ROC algorithm. Firstly, the
received signal strength decreases monotonically with the distance, so we
can conclude that a node that receives a higher signal strength is closer.
Secondly, the antennas are supposed to be isotropic. Nevertheless, the al-
gorithm is proclaimed to be resilient to irregular radio propagation patterns
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Localization in Wireless Sensor Networks 231
(x1,y1)
(x2,y2)
(x3,y3)
•
•
••©
Fig. 9.15 Ring Overlapping Circles algorithm with 3 anchors.
and capable to achieve better performance than APIT with less communi-
cation overhead [41].
Hop counter If the maximum radio range among nodes is well-
known, their distance from each other can be determined to be inferior
to that range with high probability. DV-HOP [57] algorithm uses this
connectivity measurements to determine the location of a node. All the
anchor nodes will broadcast a beacon message that will be propagated
through the network. This message includes the anchor node location and a
hop-counter that will be incremented at every hop. Each anchor node keeps
the minimum hop-counter value per anchor. This procedure enables all the
nodes in the network (including anchors) to get the shortest distance (least
number of hops) to anchors. To translate hop-count to physical distance,
an anchor i with position (xi, yi) estimates the average single hop distance
hi with the following formula:
hi =
Pp
(xi − xj)2 + (yi − yj)2
hij, (9.40)
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232 Wireless Sensor and Robot Networks
where hij is the minimum number of hops to another anchor node j with
position (xj , yj). This estimated hop size is then propagated to nearby
nodes. Finally, once the distance estimation is made to at least three an-
chors, triangulation is used to report the estimated position. The main
advantages of this algorithm are its simplicity and the fact that it does not
depend on measurement error. The more anchors can be heard, the more
precise the localization is. The main drawback is that it will only work
for isotropic networks. When an obstacle prevents an edge from appearing
in the connectivity graph the hop-counter methodology can lead to an in-
accurate location estimation. In Fig. 9.16 we can see how the number of
hops between node A and node C are equal to the hop count between node
B and node D due to the presence of an obstacle, although the later are
physically closer.
•
••
•
•
•
••
•
•
•
•
A
B
C
D
Fig. 9.16 Hop-counter with obstacle, example.
The DV-Distance algorithm is presented together with DV-hop propos-
ing a similar method but distances between neighboring nodes are used
instead of hops. Many other modifications of this algorithm to improve
performance under certain network conditions can be found in litera-
ture [65, 77].
The Amorphous algorithm [53] proposes a different approach to DV-
Hop to calculate the average single hop distance. It uses the density of the
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Localization in Wireless Sensor Networks 233
network, nlocal, to correct the average hop distance estimation, dhop, with
the help of the Kleinrock and Silvester formula [34] for a maximum radio
range R:
dhop = R
✓
1 + e−nlocal −Z 1
−1
e−nlocal
π(arccos t−t
√1−t2)dt
◆
. (9.41)
Neighborhood measurements One of the simplest coarse-grained
localization methods is using the connectivity measurement, which is more
robust to unpredictable environments, for neighbor proximity. The only
decision to make is whether a node is within the range of another. Ref-
erence nodes can be deployed through the localization area determining
non-overlapping regions. When a tracked node receives a beacon from an
anchor, it will consider that reference position as its own position.
In the case of anchors (reference positions) with overlapping regions
of coverage, Centroid Location (CL) [9] can be used. The tracked node
can listen to a given subset of anchor beacons containing their reference
positions (xi, yi) to infer its proximity to them. The node will calculate its
estimated position using the following centroid formula:
(x, y) =
✓
x1 + ...+ xN
N,y1 + ...+ yN
N
◆
. (9.42)
The same authors have also proposed a reduction of the estimation
error placing additional anchors using a novel density adaptive algorithm,
HEAP [10].
Another way to ensure a localization improvement is including weights
when averaging the coordinates of the beacon nodes. This is the Weighted
Centroid Location (WCL) [8] algorithm.
The weight is a function depending on the distance and the environment
conditions so different weights may be used. Small distances to neighboring
anchors lead to a higher weight than to remote anchors. To calculate the
approximated position of a tracked node i, every reference location j, from
the n anchor nodes in range, obtains a weight wij depending on the distance:
(xi, yi) =
Pnj=1 (wij · (xj , yj))
Pnj=1 wij
. (9.43)
To determine the associated weight to a reference either the link quality
indication (LQI) or Received Signal Strength indicator (RSSI) could be
used [7]. Nevertheless, in the LQI case, if all the references in range provide
relative high values the influence of one anchor’s LQI becomes relative low.
The Adaptative WCL (AWCL) [6] algorithm proposes to compensate high
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
234 Wireless Sensor and Robot Networks
LQI values giving more influence to the differences between the LQIs instead
of the nominal values. It reduces measured LQI values of each reference in
range by a part q of the lowest LQI (Eq. (9.44)),
(xi, yi) =
Pnj=1 ((LQIij − q ·min(LQI1...n) · (xj , yj))
Pnj=1(LQIij − q ·min(LQI1...n))
. (9.44)
A Selective Adaptive Weighed Centroid Localization (ASWCL) [19] ap-
proach has also been proposed to improve the accuracy by adapting the
weights according to their statistical distribution.
9.3.2.2 Anchor-free techniques
Anchor-based algorithms have some limitations because they need another
positioning scheme to place the beacon nodes. In some cases, the environ-
ment may prevent the use of such positioning system (e.g., GPS and indoor
locations) so pre-configured anchors providing known reference positions
are not available. In addition, the practice reveals that a large number of
beacons must be deployed to provide an acceptable positioning error [11].
They require a deployment effort and they may not scale well. In contrast,
anchor-free algorithms are able to determine each node relative coordinates
using local distance information and without relying on beacons that are
aware of their positions. Note that no absolute positions are obtained, but
this is a fundamental limitation of the problem statement and not part of
the algorithm itself. The relative coordinate space should be able to be
translated to any other global coordinate system easily. The centralized
MDS algorithm (See 9.3.1.1) is a sample of anchor-free algorithm that can
obtain final absolute positions with the help of an additional step involving
three or more beacons. Some popular distributed anchor-free approaches
are relaxation-based algorithms and coordinates stitching.
Relaxation-based algorithms These approaches are coarse grained lo-
calization methods with a refinement phase where typically each node cor-
rects its position to optimize a local error metric. We will briefly introduce
two of the most popular relaxation-based approaches.
Cooperative ranging In the cooperative ranging methodologies, ev-
ery single node plays the same role, and repeatedly and concurrently exe-
cutes the following functions:
• Receive ranging and location information from neighboring nodes.
• Solve a local localization problem.
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
Localization in Wireless Sensor Networks 235
• Transmit the obtained results to the neighboring nodes.
After some repetitive iterations the system will converge to a global
solution.
The local localization problem is revolved by making assumptions when
necessary and compensating the error through corrections and redundant
calculations as more information becomes available. These assumptions are
needed at first in order to deal with the under-determined set of equations
presented by the first few nodes. The Assumption Based Coordinate (ABC)
algorithm [72] propose the following procedure from the perspective of a
node n0:
• The node n0 is located at the position (0, 0, 0).
• The fist node to establish communication, n1, is placed at (r10, 0, 0)
where r10 is the estimated distance from some signal parameter.
• The location of the next node n2, (x2, y2, z2), is determined using the
estimated distance to both n0 and n1 and assuming that y2 > 0 and
z2 = 0,
x2 =r201
+r202
+r212
2r01
y2 =p
r202 + x22
z2 = 0.
(9.45)
• Next location n3 (x3, y3, z3) is obtained with the only assumption that
the square involved in finding z3 is positive,
x3 =r201
+r203
+r213
2r01
y3 =r203
−r223
+x2
2+y2
2−2x2x3
2y2
z3 =p
r203 + x23 + y23 .
(9.46)
From this point forth, the system of equations used to solve for further
nodes is no longer under-determined, and so the standard algorithm can be
employed for each new node. Under ideal conditions, this algorithm thus far
will produce a topologically correct map with a random orientation relative
to a global coordinate system.
The main advantage of this approach is that global resources for a cen-
tralized computing are not required. Nevertheless, the convergence of the
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
236 Wireless Sensor and Robot Networks
system may take some time and nodes with high mobility may be hard to
cover.
Spring Model The AFL (Anchor-Free Localization) [62] algorithm,
also referred to as Spring Model, describes a fully decentralized algorithm
where nodes start from a random initial coordinate assignment and converge
to a consistent solution using only local node interactions. The key idea
in AFL is fold-freedom, where nodes first configure into a topology that
resembles a scaled and unfolded version of the true configuration, and then
run a force-based relaxation procedure.
The AFL algorithm proceeds in two phases:
• The first phase is a heuristic that produces a fold-free graph embedding
which looks similar to the original embedding. Five reference nodes are
chosen, one in the center n0, and four in the periphery, n1, n2, n3 and
n4, where the couples (n1, n2) and (n3, n4) are roughly perpendicular to
each other. The choice of these nodes is performed using a hop-count
approximation to distance (e.g., the first peripheral node is selected
maximizing the number of hops to the initial node, maxh0,1). Finally
a node n5 is selected and supposed to be centered by minimizing the
distance in hops between n1 and n2 (min |h1,5−h2,5|) and the distance
between n3 and n4 (min |h3,5 − h4,5|) for contender nodes. Now, for all
nodes ni, the heuristics approximate the polar coordinates using the
maximum radio range, R, as follows:
⇢i = hi,5R
✓i = tan−1h
(h1,i−h2,i)(h3,i−h4,i)
i
.(9.47)
• The second phase uses a mass-spring based optimization to correct and
balance localized errors. It runs concurrently at each node. At any
time any node ni has a current estimated position pi that periodically
sends to its neighbors. Using these positions, the distance dij to each
neighbor nj is estimated. Also knowing the measured distance rij to
nj , a force ~Fij in the direction ~vij (unit vector from pi to pj), is given
by Eq. (9.48),~Fij = ~vij(dij − rij). (9.48)
The resultant energy Ei of node i due to the difference of the measured
and the estimated distances between nodes, can be expressed in terms
of the square of the magnitude of the forces ~Fij as Eq. (9.49),
Ei =X
j
Eij =X
j
(dij − rij)2. (9.49)
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
Localization in Wireless Sensor Networks 237
The main advantage of relaxation based algorithms is that they are
fully distributed and concurrent and they operate without anchors nodes.
Nevertheless, while the computational is modest and local, it is unclear how
these algorithms scale to much larger networks [4]. Furthermore, there are
no provable means to avoid local minima, which could be even worse at
larger scales. Traditionally, local minima have been avoided by starting the
optimization process at a favorable position, but another alternative would
be to use optimization techniques such as simulated annealing [33].
Coordinate system stitching Some methods focus on fusing the pre-
cision of centralized schemes with the computational advantages of dis-
tributed systems as we have seen in Sec. 9.3.2.2. Another approach with
the same goal that has received some attention [12, 50, 52, 57] is Coordinate
system stitching. Coordinates system stitching works as follows:
• First, it localizes clusters in the network. They normally are overlap-
ping regions composed by a single node and their one-hop neighbors.
• Then, it refines the localization of the clusters with an optional lo-
cal map for each cluster placing cluster nodes in a relative coordinate
system.
• Finally, it merges those cluster regions computing coordinate transfor-
mations between these local coordinate systems.
The fist two steps may be slightly different depending on the algorithm,
while the last third step is usually the same. In [50] sub-regions are formed
using one-hop neighbors. Then, local maps are computed by choosing three
nodes to define a relative coordinate system and using multilateration to
iteratively add additional nodes to the map, resulting in a multilateration
sub-tree.
More robust local maps can be obtained according to [52]. Instead of
using three arbitrary nodes to define a map, robust quadrilaterals are used,
considering a robust quad as a fully-connected set of four nodes where each
sub-triangle is also robust. A robust sub-triangle with a shortest side of
length b and a smallest angle θ must accomplish Eq. (9.50),
b sin2 θ > dmin, (9.50)
where dmin is a predetermined constant based on the average measured
error. The idea is that the points of a robust quad can be placed correctly
with respect to each other. Once an initial robust quad has been chosen,
any node that connects to three of the four points in the initial quad can
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
238 Wireless Sensor and Robot Networks
be added using multilateration. This preserves the probabilistic guarantees
provided by the initial robust quad, since the node form a new robust quad
with the points from the original. By induction, any number of nodes
can be added to the local map, as long as each node has a range to three
members of the map. These local maps or clusters, are now ready to be
stitched together.
Coordinates system stitching techniques are quite interesting since they
are inherently distributed and they enable the use of sophisticated local
maps algorithms. Nevertheless, registering local maps iteratively, can lead
to error propagation and perhaps unacceptable error rates as the network
grows. In addition, the algorithm may converge slowly since a single coordi-
nate system must propagate from its source to the entire network. Further-
more, these techniques are prone to leave orphan nodes because, either they
could not be added to the local map, or their local map failed to overlap
with neighboring local maps.
9.4 Other Issues in Localization
In this section we outline some aspects involved in the localization theory of
wireless ad-hoc and sensor networks that have not been covered in previous
sections such as hybrid solutions, mobility and the application of the graph
theory.
9.4.1 Graph theory and localizability
A fundamental question in the wireless sensor network (WSN) localization
is whether a solution to the localization problem is unique. The network,
with the given set of anchors, non-anchors and inter-sensor measurements, is
said to be uniquely localizable if there is a unique set of locations consistent
with the given data. Graph theory has been found to be particularly useful
for solving the above problem of unique localization. Graph theory also
forms the basics of many localization algorithms, especially for the category
of distance based localization problem, although it has been used to other
types of measurements as well.
A graphical mode for distance based localization problem can be built
by representing each sensor in the network uniquely by a vertex. An edge
exits between two devices if the distance between the corresponding sensors
is known. Note that there is always a vertex between two anchors since the
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
Localization in Wireless Sensor Networks 239
distance can be obtained form their known locations. The obtained graph
G(V,E), where V is the set of wireless communication devices and E the set
of edges, is called the underlying graph of the sensor network. Details of the
graph theoretical representations of the WSN and their use in localization
can be found in [28, 43].
9.4.2 Hybrid schemes
Hybrid schemes simply combine two or more existing techniques to achieve
a better performance such as using both multidimensional scaling (MDS)
and proximity based maps (PDS) [13]. Initially, some anchors are deployed
(primary anchors). In the first phase some sensors are selected as sec-
ondary anchors which are localized thought MDS (Sec. 9.3.1.1). Nodes
which are neither primary nor secondary are called normal sensors. In a
second phase those normal sensors are localized through proximity distance
mapping. Other examples of hybrid schemes are the use of MDS and Ad-
hoc positioning system (APS) [2] and stochastic approaches based on the
combination of deductive and inductive methods [44].
9.4.3 Mobility
Mobility of sensors nodes obviously have an impact on the localization
process. The uncertainty of the node movement may lead to increase the
difficulty of the localization task. Nevertheless, in some cases, statistical ap-
proaches having capabilities to handle uncertainty of node movements, can
tackle localization of mobile sensor nodes. The sequential Monte Carlo lo-
calization (MCL) method [27] exploits mobility to improve the accuracy and
the precision of the localization. The simultaneous localization and track-
ing scheme based on Laplace method (LaSLAT) [81] employs Bayesian
filters to accomplish the task of localizing mobile nodes, in which loca-
tion estimates are iteratively updated given batches of new measurements.
Empirical studies have shown that LaSLAT can tolerate noisy range mea-
surements and achieve satisfactory location accuracy.
The localization of static sensors using one mobile anchor equipped with
GPS has also been proposed [79]. The mobile anchor periodically transmits
a beacon message including its latest position while traversing the area
where static sensor nodes are deployed. Upon receiving the beacon packets,
a static sensor determines its location relative to the anchor according to
the received signal strength (RSS) of the beacon packet through Bayesian
August 7, 2013 16:24 World Scientific Book - 9in x 6in book
240 Wireless Sensor and Robot Networks
inference. The on beacon mobility scheduling is also subject of study [38] in
order to determine the best beacon trajectory so that each sensor receives
sufficient beacon signals with minimum delay.
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